Unitless Velocity. (Also- spatial and temporal unit questions)

[Bohandas]

Since time is a dimension, doesn't that mean that theoretically the time unit in a velocity should be able to cancel the distance unit, provided that the units themselves are of equal magnitude.

(and on that last point, is my understanding correct that a unit of time is equal in magnitude to the distance that light travels in that unit of time, or is that just something that Penrose and Light Cone diagrams do arbitrarily?)

[Rydiro]

Since time is a dimension, doesn't that mean that theoretically the time unit in a velocity should be able to cancel the distance unit, provided that the units themselves are of equal magnitude.

(and on that last point, is my understanding correct that a unit of time is equal in magnitude to the distance that light travels in that unit of time, or is that just something that Penrose and Light Cone diagrams do arbitrarily?)

Time is still a little different than space, metrics-wise. Other than that, the lightcone is as you said the most natural equivalency, i think.
No expert though.

[Anymage]

Planck units do treat certain natural values common in physics as 1, tying time units and distance units together such that c=1. That, like the other things you mentioned, is just an arbitrary convention that makes things easier on us.

You could certainly have a gigameter time unit that's equal to how long it takes light to travel that distance. You can make that up and nobody can stop you from using it. The question is if it simplifies anything, clarifies anything, or makes any sort of otherwise tricky equation easier. If none of the above apply, don't expect time-as-distance and consequent unitles velocity to take off. In our universe where time has a clear arrow while the other dimensions don't, it's not unreasonable for people to treat temporal position and motion differently.

[Radar]

You can describe time in units of length, if you want to. Technically, this will make velocity unitless, but the truth is, any physical quantity is always in units of something. In order to translate time units to length, you need some parameter that has units of velocity. Let's call it q. Your new time t' in length units will now be expressed as
t' = t*q
Given some travel distance x you get a technically unitless velocity
v' = x/t'
However, once you plug in the expression for t'
v' = x/(t*q)
it will be clear that v' is velocity in units of q even if it does not show up immediately. To see this clearly, let us introduce the regular velocity v
v = x/t
In that case we can write v' as
v' = (x/t)/q = v/q

[Jay R]

Since time is a dimension, doesn't that mean that theoretically the time unit in a velocity should be able to cancel the distance unit, provided that the units themselves are of equal magnitude.

(and on that last point, is my understanding correct that a unit of time is equal in magnitude to the distance that light travels in that unit of time, or is that just something that Penrose and Light Cone diagrams do arbitrarily?)

No. Time is a dimension. That doesn't mean it has the same units as a spatial dimension. [Except when you are making the Kessel run in under 12 parsecs, of course.]

In a few equations, the fourth dimension can be considered to be time * c (the speed of light). This has the same units as a spatial dimension, but that won't do what you want.

10 miles / second has a clear, unambiguous meaning, and no attempt at canceling units will change that.

[Chronos]

Yes. Time is a dimension, and it's perfectly reasonable to measure it in the same units as distance, making velocity dimensionless (and c = 1). It is so perfectly natural, in fact, that all actual work in relativity by actual physicists is done using such units, because it just makes things harder for no reason whatsoever to pretend that time and space must be measured in different units.

By way of analogy: Suppose that, for historical reasons, people had decided that vertical distances should always be measured in centimeters, and horizontal distances should always be measured in inches. We would observe things like that a rod's inch measurement, and its centimeter measurement, can both change, when the rod is tilted to various degrees. And we could devise formulas that described how both of these measurements change. Well, the ratio 2.54 cm/inch would show up all over the place in these formulas. What is that quantity? Well, it's a vertical measurement divided by a horizontal measurement, which means it's a slope. We'd call it the Slope of... Well, the Slope of Something. We'd find some slope phenomenon that had that slope, and refer to that constant in terms of it. But what, actually, is that quantity? Really, it's just 1: 2.54 cm IS one inch, and so 2.54 cm/inch is just 1, and it's only because of historical quirk that it shows up as anything other than 1.

10 miles / second has a clear, unambiguous meaning, and no attempt at canceling units will change that.

Yes, it means a speed of 5.37e-5.

[Radar]

Yes. Time is a dimension, and it's perfectly reasonable to measure it in the same units as distance, making velocity dimensionless (and c = 1). It is so perfectly natural, in fact, that all actual work in relativity by actual physicists is done using such units, because it just makes things harder for no reason whatsoever to pretend that time and space must be measured in different units.

Moreover, the time direction is not distinct - it depends on the observer, so any separation between space and time is just a local point of view and not a global quality of the universe.

Yes, it means a speed of 5.37e-5.

In units of c. Formal lack of units does not mean there are no units at all.

[Chronos]

Sure, "in units of c". But c is unitless and is equal to 1. Not all physical quantities need to have units. In fact, all of the truly fundamental quantities are unitless.

[Radar]

Sure, "in units of c". But c is unitless and is equal to 1. Not all physical quantities need to have units. In fact, all of the truly fundamental quantities are unitless.

Technically yes, but there is an important point to be made that even if you switch to unitless velocity, you still need to have a reference value to understand, what a speed of - for example - 0.1 means.

[KillianHawkeye]

This might actually be the strangest conversation I've read on this site....

[Chronos]

Quoth Radar:

Technically yes, but there is an important point to be made that even if you switch to unitless velocity, you still need to have a reference value to understand, what a speed of - for example - 0.1 means.

If your reference is anything other than c, then yes, you need to specify.

It's as if you were specifying the slope of a hill. You could say that a slope is 0.1 times as steep as some specified reference slope. But you can also say that the slope is just 0.1 . If you don't specify any other reference, then that means that the vertical rise is 0.1 times the horizontal run.

[asda fasda]

If your reference is anything other than c, then yes, you need to specify.

It's as if you were specifying the slope of a hill. You could say that a slope is 0.1 times as steep as some specified reference slope. But you can also say that the slope is just 0.1 . If you don't specify any other reference, then that means that the vertical rise is 0.1 times the horizontal run.

I think that you just implying the unit in this sentence (both sentences, about the slope and velocity). It's not that there is no unit, it is that you assume that when you not define the you unit people will be able correctly guess which one you are using.

Units are for describing the set from which particular number are taken from, to show that they are different. For example having velocity and slope both equal 0,1 (of there respective units) doesn't make them equal as they are describing different things. In physics you are thought to write down your units in order to make sure that at some point you will not subtract the time from distance as this would be wrong. Moreover as some items are connected to each other this lets you use transformation on units to do the math on those items. But units are always there.

Look at this as every unit is just a different axis on a graph, you can write position on this graph as (2,3) but only because we all agreed that notation is (x,y). Those number are from different sets and you cannot do any arithmetic's on those number so they have different unit (x and y respectfully) .

[Radar]

I think that you just implying the unit in this sentence (both sentences, about the slope and velocity). It's not that there is no unit, it is that you assume that when you not define the you unit people will be able correctly guess which one you are using.

Units are for describing the set from which particular number are taken from, to show that they are different. For example having velocity and slope both equal 0,1 (of there respective units) doesn't make them equal as they are describing different things. In physics you are thought to write down your units in order to make sure that at some point you will not subtract the time from distance as this would be wrong. Moreover as some items are connected to each other this lets you use transformation on units to do the math on those items. But units are always there.

Look at this as every unit is just a different axis on a graph, you can write position on this graph as (2,3) but only because we all agreed that notation is (x,y). Those number are from different sets and you cannot do any arithmetic's on those number so they have different unit (x and y respectfully) .

Everything I wanted to say in a nicely comprehensive form.

Physics is a map between reality and mathematical models. Units are a map between measurable quantities and numbers.

[Fat Rooster]

Everything I wanted to say in a nicely comprehensive form.

Physics is a map between reality and mathematical models. Units are a map between measurable quantities and numbers.

This is true, but some things are truly unitless in science; usually ratios*. When you divide something measured in metres by something else measured in metres the unit disappears. It is not that you have forgotten the unit, the unit is 'unitless'. It's value is independent of the system of units we are using to measure the system.

*Damping factors in oscillators are a good one to think about. You can measure the magnitude of the second peak compared to the first one and the result is a ratio with physical meaning. This ratio is the same no matter whether you are measuring in metres, inches, parsecs, light seconds, or poundals/BTU. Normalised directions are too, because you have a direction (measured as a vector of metres say) divided component wise by it's magnitude (measured in metres), giving a vector of unitless values. The example quoted was gradients of slopes, which are also clearly measured by taking a distance and dividing it by another distance, meaning the unit disappears (or becomes a unitless constant).

Relativity gives us a consistent model in which we can use the same units for both measurement of distance and of time. When we measure velocity we then get a ratio of two values with the same unit, meaning the result is unitless. Velocities in this system behave like to normalised directions.

[Rakaydos]

https://physics.nist.gov/cuu/Units/current.html

Unit of length meter The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s-1, where the second is defined in terms of ΔνCs.

Unit of mass kilogram The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10-34 when expressed in the unit J s, which is equal to kg m2 s-1, where the meter and the second are defined in terms of c and ΔνCs.

Unit of time second The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the cesium frequency ΔνCs, the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s-1


Unit of electric current ampere The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 x 10-19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ΔνCs.

Unit of thermodynamic temperature kelvin The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 649 x 10-23 when expressed in the unit J K-1, which is equal to kg m2 s-2 K-1, where the kilogram, meter and second are defined in terms of h, c and ΔνCs.

Unit of amount of substance mole The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76 x 1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol-1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.

Unit of luminous intensity candela The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 x 1012 Hz, Kcd, to be 683 when expressed in the unit lm W-1, which is equal to cd sr W-1, or cd sr kg-1 m-2 s3, where the kilogram, meter and second are defined in terms of h, c and ΔνCs.

Science has put a lot of work into redefining our everyday units into something that fits "natural" measurements.

You could round the meter to "1/300,000,000 of a second" and be off by less than a milimeter.

[Bohandas]

This might actually be the strangest conversation I've read on this site....

See also the other thread I just started:

https://forums.giantitp.com/showthre...1#post25095181